# [FOM] Eliminating AC

Craig Smorynski smorynski at sbcglobal.net
Thu Mar 21 16:11:06 EDT 2013

```Maybe you can wave your hands with "hereditarily ordinally definable sets" and point out that ordinal definability makes every set well-orderable (whence AC is true) and the set of natural numbers is in HOD.

On Mar 21, 2013, at 1:47 PM, Joe Shipman wrote:

> I am looking for something that I can explain in a few minutes to a mathematician who is unfamiliar with Godel constructibility, so that he will feel he understands why the result is true.
>
> -- JS
>
> Sent from my iPhone
>
> On Mar 21, 2013, at 12:17 AM, Craig Smorynski <smorynski at sbcglobal.net> wrote:
>
> The original proof by noticing that the natural numbers in V and L are the same must surely be as simple as possible, which of course is not to say that it is simple.
>
> On Mar 20, 2013, at 7:52 PM, Joe Shipman wrote:
>
>> What is the simplest way to see that any arithmetical consequence of ZFC is a consequence of ZF?
>>
>> -- JS
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> Craig
>
>
>
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Craig

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